| Protein play a crucial role in life's processes. In recent years, there has been a growing interest in exploiting the relationship between molecular structure and biological activity. Many scientists from different research areas, e.g. biologist, chemist, computational scientist, mathematician or physicist, are trying to make some progresses from the view of either experiments or computational analysis. Now one of the main themes is using well-known protein structure to predict unknown protein molecular activity, based on the knowledge of computational science, biological informatics, computational biology, computational chemistry and so on.Existing molecular models are mostly constructed by atom positions and connections between atoms. They don't contain the information of molecular mechanics or molecular force energy, and may limit the study of molecular structure-activity relationships. In this paper, we consider the protein as a whole system composed of different force interactions, which is called "Molecular Field". Then we use Level Set evolutional surfaces to extract and analyze the feature of molecular field, and show the results by several kinds of rendering techniques. Our main contributions include:Molecular Field in a quantitative manner is constructed; Then by applying the first order and the second order local differential operators on individual node, a set of critical points which potentially depicts the active region of protein molecule are found; Also this chapter gives some results about computing a sequence of molecular potential energy in the data field and interactively exploring the potential "tunnel" region exhibiting biological sense. In addition, the point-based, surface and volume rendering techniques are exploited to find the macro-structure inside the data field.Level Set model in Molecular Field is presented, which is based on Chan-Vese model but doesn't need to re-initialize Level Set function in period; By using the geometrical measure, the convergence of Level Set model is detected; And the influences of each parameters in the whole process are discussed; Finally this chapter shows results and discussion about the Level Set models on HIV-1 protease Molecular Field and DPS protein Molecular Fields. Techniques of topological analysis and multi-attribute comparison of Level Set evolutional surfaces are presented. On one side, we propose a volume feature extraction function to detect the topological features imbedded in the Level Set surfaces; On the other side, we compute the geometrical and biological attributes on Level Set surface, and explore multiple attributes on the Level Set surface to study the progression of the Molecular Field.Based conformal geometry theory, the further study of geometrical and topological feature on Level Set surface is presented. First, we compute the gaussian curvature and mean curvature of Level Set surface on Molecular Field, to explore the surface local geometrical feature distribution; Also, by using Euler-characteristic formular, the global surface topological feature is described by genus number; Then we classify surfaces by the constant curvature measure, and conformal map the Level Set surfaces on Molecular Field; Finally, we compute the conformal factor distribution on Level Set surface, which indicates the surface intrinsic feature. Then we compare different molecular field using their conformal factors.Our approach provides an intuitive way to study molecular structure-activity relationships. Initial results on some typical protein molecules reveals their active region inside, such as the escape route of water molecules hidden in the HIV-1 protease, the internal cavity of the DPS protein for the iron atom's entry or deposition, the molecular activity change of HIV-1 protease in water during SMD process, and the Globin homology protein group similarity during evolution can be successfully found, which are in accordance with biological experiments.
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